Hey there!
Yes, as you have a Time Machine backup, you can revert to Catalina with all your data at the time the backup was done, also if you have a current backup, you can restore any files that were added to your Mac from the current backup. As well, even if you erase your Mac as new with Catalina, you can simply restore your iCloud data by signing into your account after the installation.
To restore from your Catalina backup:
Boot your Mac to recovery mode. Startup holding Command R.
Select the option to Restore from a Time Machine backup. You’ll be prompted to erase your disk beforehand, this is the only way to revert the OS using Time Machine, then select a backup from the date it had Catalina.
Likewise, if you need to, you can create a Bootable Installer with Catalina, boot from that, erase your HD using Disk Utility, then Reinstall Mac OS Catalina using recovery mode:
Restore your Mac from a backup
https://support.apple.com/en-us/HT203981
How to create a bootable installer for macOS
https://support.apple.com/https://support.apple.com/guide/mac-help/change-your-mac-startup-disk-mchlp1034/machelp/change-your-mac-startup-disk-mchlp1034/mac
(This is to start up your Mac from the bootable installer).
As dialabrain mentioned, you can see the OS install dates in System Information. If you need to, you can also open your external drive in Finder. Open backups.backup.db > (Name of your Mac), and visibly see all the dates of your backups to ensure they are there.
Likewise, for this (these) apps that aren’t working as they should. Typically Time Machine backs up their data, it’s just a matter of where it’s stored on your HD, but if so, can usually be found.
Although sometimes that’s not the case, and the data is stored in a cloud somewhere, if iCloud, that’s easy to recover, if elsewhere, I wouldn’t know. However if the third party apps aren’t working as they should after Big Sur, I’d say, (If available, I’d check them for any updates, or if need be, await an update to work with Big Sur), sometimes it takes a while for others to catch up. Usually on the developer’s website they’ll have more information.
Hope all works out!
